The [[Laplacian]] [[Digital Image|Image]] [[Signal Filter|Filter]] is a filter for finding the second order [[Partial Derivative|partial derivative]] of some image $f$.
$
\huge \begin{align}
\nabla^{2}f(x,y) &= \frac{\partial^{2}f}{\partial x^{2}} + \frac{\partial^{2}f}{\partial y^{2}} \\
&= 2f(x,y)-f(x-1,y)-f(x+1,y) \\
&+ 2f(x,y)-f(x,y-1) - f(x,y+1)
\end{align}
$
The [[Convolution|Kernel]] is:
$\huge \begin{align}
\mat{
0&1&0\\
1&-4&1\\
0&1&0
} \end{align} $
For a $N_{8}$ (8 neighbor) laplacian,
$\huge \begin{align}
\begin{bmatrix}
1 & 1 & & 1 \\
1 & -8 & 1 \\
1 & 1 & 1
\end{bmatrix}
\end{align} $